Question Video: Finding the Length of the Altitude in a Right Triangle given the Triangle’s Dimensions Mathematics

Video Transcript

Determine the length of line segment 𝐴𝐷.

We note that 𝐷 is the projection of 𝐴 onto the line 𝐵𝐶 and that the triangle 𝐴𝐵𝐶 is a right triangle at 𝐴. And we recall that the corollary to the Euclidean theorem, that is, the altitude rule, tells us that 𝐴𝐷 squared is equal to 𝐵𝐷 multiplied by 𝐶𝐷. This tells us that the altitude or height of the right triangle squared is the product of the lengths of the segments of the hypotenuse when it is split by the altitude.

We’re given that the two segments are 2.5 centimeters, that’s 𝐶𝐷, and 6.4 centimeters, that’s 𝐵𝐷. And substituting these values in, we have 𝐴𝐷 squared is 6.4 multiplied by 2.5. This evaluates to 16, so 𝐴𝐷 squared is 16. Now, taking the square root on both sides of the equation, noting that 𝐴𝐷 is a length and so is nonnegative, we get 𝐴𝐷 is the square root of 16, which is four. The length of 𝐴𝐷 is therefore four centimeters.